Fund Project:
This research is partly supported by the National Institute of Information and Communications Technology, Japan, and by the Japan Society for the Promotion of Science Grant Nos. 23246071 and 26289116, and the Villum Foundation through their VELUX Visiting Professor Programme 2013-2014

The first construction of strongly secure quantum ramp secret sharing by Zhang and Matsumoto had an undesirable feature that the dimension of quantum shares must be larger than the number of shares. By using algebraic curves over finite fields, we propose a new construction in which the number of shares can become arbitrarily large for fixed dimension of shares.

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H. Yamamoto, Secret sharing system using (k; l; n) threshold scheme, Electronics and Communications in Japan (Part I: Communications), 69 (1986), 46-54, (the original Japanese
version published in 1985)
doi: 10.1002/ecja.4410690906.

Daniele Bartoli, Leo Storme.
On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric.
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doi: 10.3934/amc.2014.8.271